On some parameters of Parity Signed Graphs
Reshma Ra , Gayathri Hb, and Supriya Rajendran c
a ,b,c,Department of Mathematics, Amrita School of Arts and Sciences, Amrita Vishwa Vidyapeetham, Kochi
Article History: Received: 11 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published
online: 4 June 2021
Abstract: The parameters of parity signed graph mentioned in this paper are- rna and adhika number. The rna number of a
parity signed graph S∗ is the minimum number of negative edges among all possible parity labelling of it’s underlying graph G, whereas adhika number is the maximum number of positive edges among all possible parity labelling of it’s underlying graph G.This paper mainly focuses on rna number and adhika number for certain classes of parity signed graphs..
Keywords: Signed Graph, Parity Signed Graph, rna number, adhika number . AMS subject classification:05C78, 05C22
1. Introduction
Graph Theory deals with the study of graphs. A graph G= (VG, EG, RG) is an ordered triplet where,
VG: vertex set of G,
EG: edge set of G and
RG: function which is defined from set EG to an unordered pair of (same/distinct) vertices of VG which are
called its endpoints.
We consider graphs with atmost one edge between every pair of vertices. For terminologies related to graph we refer to [1,2]. Also ⌊𝑛⌋ denotes the greatest integer less than or equal to n and ⌈n⌉ denotes the lowest integer greater than or equal to n.
A Signed graph is a special kind of graph, where each edge receives either a positive sign or negative sign. The idea of Signed graphs has been first introduced by Frank Harary in [3] and this concept was used by Frank Harary and Dorwin Cartwright to handle a problem in social psychology. We represent signed graph as 𝑆∗= (𝐺, 𝛼) where G = (VG, EG) is called underlying graph of 𝑆∗and α: 𝐸𝐺→ {+, −} is called the signature of 𝑆∗ . The set 𝐸+(𝑆∗) indicates set of positive edges (i.e, edges receiving positive sign). The set 𝐸−(𝑆∗) indicates set of negative edges(i.e, edges receiving negative sign). An all-positive signed graph is a signed graph in which there are no negative edges, whereas all-negative signed graph is a signed graph in which there are no positive edges. If a Signed graph 𝑆∗ is either all- positive or all-negative then it is homogeneous, otherwise it is heterogeneous. For a
signed graph S∗, if we take product of edge signs around every cycle and if we get positive, then we say that
signed graph 𝑆∗ is balanced, otherwise it is unbalanced.
Signed graph has applications in industrial as well as theoretical fields. It has most of it’s applications in the field of social networking i.e., they were mostly used for demonstrating social scenario among group of people where vertices represent people, positive edges represent friendship and negative edges represents enmities between them. We refer [7] for signed graphs. To know more detailed information in signed graphs, refer to [8].
M. Acharya et al. in [4] has introduced the concept of parity signed graph. In this article, they initiated the study on parity labelling in signed graphs and also found rna number for some classes of parity signed graphs. M. Acharya et al. in [5] has given fundamental descriptions for graphs like- parity signed stars, bistars, cycles, paths and complete bipartite graphs and also they found rna number for few parity signed graphs. Athira et al. in [6] has initiated the research on sum signed graphs and they also established rna number on some graph classifications and presented some of the features of sum signed graphs. T. Zaslavsky in [7] initiated the conception of matroids of signed graphs, which generalise both polygon matroids and even- circle matroids of ordinary graphs. Debanjan et al. in [9] has considered different problems like representation of fair friendship, representation of unfair friendship, educational field problems like mark distribution of some students in certain subjects, industrial problems like which machine is performing badly so that it can be replaced by better ones, medical problems like study of spreading of cancer and all these were studied with the help of signed graphs.
Consider a graph G and a function 𝛿: 𝑉𝐺→ {1,2, … … , 𝑛}. Let α : EG → {+, −} be a function in order that for
any edge xy in G, 𝛼(𝑥𝑦) = + , if 𝛿(𝑥) and 𝛿(𝑦) are of same parity (i.e, both end vertices should be either odd or even) and 𝛼(𝑥𝑦) = − , if 𝛿(𝑥) and 𝛿(𝑦) are of opposite parity. 𝑆∗= (𝐺, 𝛼) is parity signed graph if δ is bijective. Now we define two parameters of parity signed graph - rna number and adhika number.
The Sanskrit word for ‘-’ is ‘rna’ which implies debt. The term ‘rna’ is pronounced as rina where ri is same as ri in ribbon and na is same as na in corona. The rna number of a parity signed graph S∗ is the minimum
number of negative edges among all possible parity labelling of it’s underlying graph G, which is indicated by the symbol 𝛼−(𝑆∗). The main application of rna number is mainly found in sociology. ‘rna’ number gives us the slightest amount of discomfort among a group of people i.e, rna number is directly proportional to amount of discomfort. In other words, when the value of rna number is small, the discomfort among people is low. Whereas, adhika number is the maximum number of positive edges among all possible parity labelling of it’s underlying graph G, which is indicated by the symbol 𝛼−(𝑆∗) . We can also define adhika number as shown below,
𝛼+(𝑆∗)= Number of edges - 𝛼−(𝑆∗)
The following inequalities are true for a parity signed graph 𝑆∗ [4].
𝛼−(𝑆∗)≤| 𝐸−(𝑆∗) | and | 𝐸+(𝑆∗)|≤ 𝛼+(𝑆∗). We recall few results on rna number of parity signed graphs.
Proposition 1. [4]
If G≅ Pm,α−(Pm) = 1 , where Pm is the path on m vertices .
Proposition 2. [4]
If G≅ Cn,α−(Cn) = 2 , where Cnis the cycle on n vertices .
Proposition 3. [4]
If G ≅ K1,m , α−(K1,m) = ⌈ m
2⌉ , where K1,m is the star graph on (m+1) vertices.
Proposition 4. [4]
If G ≅ Km , α−(Km) = ⌊ m
2⌋ ⌈ m
2⌉, where Km is the complete graph on m vertices and m ≥ 2.
Proposition 5. [4]
Let Z be a parity signed tree. Then α−(Z) = |E−(Z)| iff Z ≅ K1,n , where n ϵ N is odd.
Proposition 6. [5]
If G ≅ Wm , α−(Wm) = ⌈ m+4
2 ⌉ , where Wm is the wheel graph on m vertices.
Proposition 7. [5] Let G ≅ Pn( or Cn). The rna number of Pn , where n ≥ 2 is 1 . The rna number of Cn
is 2 . Also , α−(G) = α+(G) iff G ≅ P
3(or C4).
Proposition 8. [5] There exists a parity signed graph S∗ such that α−(S∗)= k, where k .
Proposition 9. [5] If S∗is a parity signed graph such that
α−(S∗) = 1 , then S∗ has a cut-edge.
Proposition 10. [5] If S∗ is a parity signed graph having a cut-edge which joins two graphs having orders
which differ by at most one, then α−(S∗) = 1. .
In this paper, we found rna number and adhika number for some parity signed graph classes whose underlying graphs are Complete Bipartite graph, Centipede graph, Barbell graph, Fan graph, Helm graph, Sunlet graph and Friendship graph.
rna number and adhika number
2.Complete Bipartite Graph [10]
A complete bipartite graph Kp,q where p,q > 0 is a graph whose vertex set can be partitioned into two disjoint sets in such a way that every vertices of the first set are adjacent to every vertices of the second set. This graph has total (p + q) vertices and pq edges.
α−(K
p,q) = ⌈ pq
2⌉
Proof. Let Kp,q be a complete bipartite graph. Let the first vertex set be X = {vi : 1 ≤ i ≤ p} and the other
vertex set be
Y = {𝑢𝑖 : 1 ≤ i ≤ q}.
Let 𝛾 ∶ 𝑉(𝐾𝑝,𝑞) → {1,2,3, … . . (𝑝 + 𝑞)} be the vertex labelling function. Let us label the vertices v1,v2 ...,vp of
the first set X with integers consecutively and then vertices u1,u2 ...,uq of the second set with remaining integers
consecutively. Then we get 2 cases: Case 1 : pq is even
In this case, (𝑝𝑞
2) pairs of end vertices are of different parity. Hence there will be ( 𝑝𝑞 2) negative edges. Therefore, . Case 2 : pq is odd In this case, (𝑝𝑞+1
2 ) pairs of end vertices are of different parity. Hence,there will be ( 𝑝𝑞+1
2 ) negative edges. Therefore,
Thus we can conclude, .
Corollary 1.
For any complete bipartite graph Kp,q where p,q > 0
3.. Centipede Graph [10]
m-Centipede graph Cm where m ≥ 1 is a graph which is obtained by connecting one pendant edge to every
vertex of the path Pm. This graph has total 2m vertices and 2m-1 edges.
Theorem 2. If G ≅ 𝐶𝑚 with m > 1,
𝛼−(𝐶𝑚) = {
1 , 𝑖𝑓 𝑚 𝑖𝑠 𝑒𝑣𝑒𝑛 2 , 𝑖𝑓 𝑚 𝑖𝑠 𝑜𝑑𝑑 Proof.
Let Cm be m-centipede graph. Let the pendant vertices be
{vi : 1 ≤ i ≤ m} and let the vertices of the path be {ui : 1 ≤ i ≤ m}. Let f :V (Cm) → {1,2...2m} be the vertex
labelling function. Then we have m vertices with odd labels and m vertices with even labels. Case 1 : m is even
For 1 ≤ i ≤ m
2 label vi,ui’s with odd integers and for ( m
2) + 1 ≤ i ≤ m , label vi,, ui’s with even integers. From this labelling one edge in the path having end vertices
of opposite parity receives negative sign. Therefore, α−(C
m) = 1.. Case 2 : m is odd.
For 1 ≤ i ≤ ⌊m
2 ⌋, label vertices vi,ui’s with odd integers and for ⌈ m
2⌉ + 1 ≤ i ≤ m, label vertices vi,ui’s with even integers. Then there will be one pendant edge in middle, label that viui with remaining integers of different
parity. From this labelling one edge in path and one pendant edge have end vertices of different parity. Hence they both will receive negative sign. Therefore,
α−(C m) = 2 Thus we conclude that
α−(C
m) = {1 , if m is even2 , if m is odd
Corollary 2.
For any m-centipede graph Cm where m > 1,
𝛂+(𝐂 𝐦) = {
(2m − 1) − 1 = 2m − 2 , if m is even (2m − 1) − 2 = 2m − 3 , if m is odd
4.Barbell Graph [11]
t-Barbell graph Bt where t ≥ 2 is the simple graph which we get by joining two replicas of Kt with a bridge,
where Kt denotes complete graph on t vertices. This graph has total 2t vertices and 2 .(t2) + 1 edges.
Theorem 3.
If G ≅ Bt with t ≥ 2,then α−(Bt) = 1.
Proof. Let Bt be t-Barbell graph. Let X = {ui : 1 ≤ i ≤ t} be the vertices of first replica of Kt and let Y = {vi : 1 ≤
i ≤ t} be the vertices of second replica of Kt. Let u1v1 be the bridge connecting two replica’s of Kt.
Let f : V (Bt) → {1,2,...,2t} be the vertex labelling function. Label the vertices of two sets with function f(ui) =
2i − 1 and f(vi) = 2i, where i=1,2,...,t.
Then we get, the bridge u1v1 which connects the two replica’s of K t have end vertices of different parity and
all other edges have end vertices of same parity. Thus it is clear that only one edge receives negative sign, which is bridge. Therefore,
α−(B
t) = 1.
Corollary 3. For any t-Barbell graph where t ≥ 2,
Fan Graph [10]
A Fan graph fq where q ≥ 2 is obtained by joining all vertices of Pq (path graph on q vertices) to a common vertex called center. A Fan graph has total (q+1) vertices and (2q-1) edges.
Theorem 4. If G ≅ fq , q ≥ 2 then, α−(f 𝐪) = { q+2 2 , if q is even q+3 2 , if q is odd Proof. Let fq be a fan graph. Let the vertices of path be {vi: 1 ≤ i ≤ q}. Let the central vertex be c.
Let δ : V (fq) → {1,2...,q + 1} be the vertex labelling function such that δ(c) = 1. Case 1 : q is even.
In path Pq , label first q
2 vertices with odd integers and the remaining q
2 vertices with even integers. Then we get, q
2 edges connecting central vertex receive negative sign. In addition to that, one middle edge of Pq also receives negative sign. Hence, (
q
2) + 1 = ( q+2
2 ) edges receives negative sign. Therefore,
Case 2 : q is odd. In path Pq , label first ⌊
q
2⌋ vertices with odd integers and the remaining ⌈ q
2⌉ vertices with even integers Then we get (q+1
2 ) edges connecting central vertex and one edge in Pq have end vertices of opposite parity. Hence (q+1
2 ) + 1 = ( q+3
2 ) edges receives negative sign. Therefore,
. Thus, we can conclude that
α−(f 𝐪) = { q+2 2 , if q is even q+3 2 , if q is odd
Corollary 4. For any fan graph fq , where q ≥ 2,
α+(f q) = { (2q − 1) − q+2 2 = 3q−4 2 , if q is even (2q − 1) −q+3 2 = 3q−5 2 , if q is odd 5.Helm Graph [10]
Helm Graph Hm where m ≥ 3 is a graph which is obtained from Wm, by attaching a single pendant edge to every vertex of the cycle in Wm, where Wm denotes wheel on m vertices. This graph has total 2m+1 vertices and 3m edges.
Theorem 5. If G ≅ Hmwhere m ≥ 3,
α−(H
m) = m.
Proof. Let Hm be a Helm graph. Let the hub (central vertex) be c. Let the pendant vertices be {vi: 1 ≤ i ≤ m} and let the vertices of the cycle be {ui: 1 ≤ i ≤ m}.
Let γ: V(Hm) → {1,2, … . , (2m + 1)} be the vertex labelling function such that γ(c) = 1.
First we label the vertices of the cycle ui with odd integers and then label the pendant vertices viwith even integers. From this labelling, all the m pendant edges receives negative sign. Therefore,
α−(H
m) = m.
Corollary 5. For any Helm graph Hm where m ≥ 3,
α+(H
m) = (3m) − m = 2m .
6.Sunlet Graph [10]
p-Sunlet graph Sp where p ≥ 3 is a graph which we get by adding pendant edges to every vertices of cycle
graph Cp. This graph has total 2p vertices and 2p edges.
Theorem 6. If G ≅ SP where p ≤ 3,
α−(S p) = {
3, if p is odd 2, if p is even
Proof. Let SP be p-Sunlet graph. Let the pendant vertices be {vi: 1 ≤ i ≤ p} and let vertices of cycle be {ui: 1 ≤ i ≤ p}.
Let β: V(Sp) → {1,2, … … . ,2p} be the vertex labelling function. Then we have p vertices with odd labels and p vertices with even labels.
We label the (p − 1) pair of viui with integers of same parity. Then we get one pair of viui and two edges of cycle have end vertices of different parity. Hence, these three edges receives negative sign. Therefore, α−(S
p) =
3 . Case 2: p is even
We label the p pair of viui with integers of same parity. Then we get two edges of cycle having end vertices of different parity and hence these two edges receives negative sign. Therefore, α−(S
p) = 2 . Hence, we can conclude that
α−(S p) = {
3, if p is odd
2, if p is even
Corollary 6. For any p-Sunlet graph Sp where p ≥ 3
α+(S p) = {
(2p) − 3, if p is odd (2p) − 2, if p is even
7.Friendship Graph [10]
Friendship graph Fmwhere m ≥ 2 is a planar undirected graph which is obtained by connecting m identicals of cycle graph C3 to a common vertex. It has total (2m+1) vertices and 3m edges.
Theorem 7. If G ≅ Fm where m ≥ 2, then
α−(F
m) = {m + 1, if m is oddm, if m is even
Proof. Let central vertex of Fm be c. Let viui where i = 1,2, … … . . , mdenote the edges for each cycle of Fm.
Let ρ: V(Fm) → {1,2, … . . ,2m + 1} be the vertex labelling function such that ρ(c) = 1. Then m vertices recieves even labels and (m+1) vertices receives odd labels.
Case 1: m is even
Label the vertices vi, ui of each cycle with integers of same parity. Then m edges connecting central vertex c with even label vertices receives negative sign. Therefore, α−(F
m) = m Case 2: m is odd
Label the vertices vi, ui of each cycle in clockwise direction as follows. For 1 ≤ i ≤ ⌊m
2⌋, label vertices viui′s with odd integers and for ⌈m
2⌉ + 1 ≤ i ≤ m
,label vertices viui′s with even integers. Then there will be one edge viui of a cycle, label that viui with remaining integers. From this labelling, m edges connecting central vertex c with even label vertices and one edge viui of a cycle have end vertices of different parity and hence, receives negative sign.
Therefore,
α−(F
m) = m + 1 Thus we conclude that,
α−(F
m) = {m + 1, if m is oddm, if m is even
Corollary 7. For any Friendship graph Fm where m ≥ 2,
α+(F m) = {
(3m) − m = 2m, if m is even (3m) − (m + 1) = 2m − 1, if m is odd
8..Conclusion
In this paper, we found rna number and adhika number for certain classes of parity signed graphs whose underlying graphs are Complete Bipartite Graph, Centipede Graph, Barbell Graph, Fan Graph, Helm Graph,
Sunlet Graph and Friendship Graph. We can study the parameters of derived signed graphs of these graph classes.It can be further extended to graph operations of signed graphs
References
1. F. Harary, Graph theory, Addison-Wesley, Reading, Mass., 1969. 2. D. B. West, Introduction to Graph Theory, Prentice-Hall of India,1999.
3. F. Harary, On the notion of balance of a signed graph, Michigan Math. J., 2 (6) (1953), 143–146.
4. M. Acharya and J.V.Kureethara, Parity Labeling in Signed Graphs, J. Prime Research in Math., 5 (2009), 165–170.
5. M.Acharya, J.V. Kureethara, and T.Zaslavsky, Characterizations of Some Parity Signed Graphs, arXiv preprint arXiv:2006.03584 (2020).
6. Athira P Ranjith, J.V. Kureethara, Sum signed graphs - I,AIP Conference Proceedings,030047 (2020) 7. T. Zaslavsky, Signed graphs, Discrete Appl. Math,. 4(1) 47–74(1982)
8. T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin., Dynamic survey DS#6, 2005.
9. Debanjan Banerjee and Anitha Pal , Applications of Parity Signed Graph in decision making, International Journal Of Computers & Technology ,Vol.14, No.3 (2015)
10. Joseph A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 18, (2015), #DS6.
11. Ghosh, S. Boyd and A. Saberi, Minimizing effective resistance of a graph, Proc.17th Internat. Sympos. Math. Th. Network and Systems,(2006), 1185-1196.
